A transference principle for involution-invariant functional Hilbert spaces

Abstract

Let $\sigma :{\mathbb{C}}^d\to {\mathbb{C}}^d$ be an affine-linear involution such that $J_{\sigma }=-1$ and let $U,V$ be two domains in ${\mathbb{C}}^d.$ Let $\varphi :U\to V$ be a $\sigma$-invariant $2$-proper map such that $J_{\varphi }$ is affine-linear and let $\mathcal{H}(U)$ be a $\sigma$-invariant reproducing kernel Hilbert space of complex-valued holomorphic functions on $U.$ It is shown that the space ${\mathcal{H}}_{\varphi }(V):=\{f\in \mathrm{Hol}(V):J_{\varphi }\cdot f\circ \varphi \in \mathcal{H}(U)\}$ endowed with the norm $\Vert f{\Vert }_{\varphi }:=\Vert J_{\varphi }\cdot f\circ \varphi {\Vert }_{\mathcal{H}(U)}$ is a reproducing kernel Hilbert space and the linear mapping ${\Gamma }_{\varphi }$ defined by ${\Gamma }_{\varphi }(f)=J_{\varphi }\cdot f\circ \varphi ,$ $f\in \mathrm{Hol}(V),$ is a unitary from ${\mathcal{H}}_{\varphi }(V)$ onto $\{f\in \mathcal{H}(U):f=-f\circ \sigma \}.$ Moreover, a neat formula for the reproducing kernel ${\kappa }_{\varphi }$ of ${\mathcal{H}}_{\varphi }(V)$ in terms of the reproducing kernel of $\mathcal{H}(U)$ is given. The above scheme is applicable to symmetrized bidisc, tetrablock, $d$-dimensional fat Hartogs triangle and $d$-dimensional egg domain. Although some of these are known, this allows us to obtain an analog of von Neumann's inequality for contractive tuples naturally associated with these domains.

arXiv commentMajor revision